Problem: Factor the following expression: $-4$ $x^2+$ $5$ $x+$ $21$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-4)}{(21)} &=& -84 \\ {a} + {b} &=& & & {5} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-84$ and add them together. Remember, since $-84$ is negative, one of the factors must be negative. The factors that add up to ${5}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${12}$ $ \begin{eqnarray} {ab} &=& ({-7})({12}) &=& -84 \\ {a} + {b} &=& {-7} + {12} &=& 5 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-4}x^2 {-7}x +{12}x +{21} $ Group the terms so that there is a common factor in each group: $ ({-4}x^2 {-7}x) + ({12}x +{21}) $ Factor out the common factors: $ x(-4x - 7) - 3(-4x - 7) $ Notice how $(-4x - 7)$ has become a common factor. Factor this out to find the answer. $(-4x - 7)(x - 3)$